Quantum Fluid of Microwave Photons

• Quantum fluid of microwave photons is a propagating, interacting many-body system in superconducting circuits that mimics ultracold atomic gases.
• Its dynamics are governed by the Lieb–Liniger model with tunable photon–photon interactions, allowing coherent conversion to a fermionized Tonks–Girardeau state.
• Adiabatic tapering of Josephson-junction transmission lines facilitates experimental access to key observables like antibunching and Friedel oscillations.

A quantum fluid of microwave photons is a propagating, interacting photonic many-body system realized in superconducting circuitry, where microwave photons display collective quantum phenomena analogous to those of ultracold atomic gases. In state-of-the-art superconducting Josephson-junction (JJ) transmission lines, the intrinsic circuit nonlinearity mediates photon–photon interactions, enabling a regime where the microwave photon field evolves analogously to a one-dimensional Bose gas governed by the Lieb–Liniger model. The degree of photon–photon interaction, and thus the quantum properties of the propagating field, are tunable by circuit parameters and drive conditions. Adiabatic spatial variation of the JJ transmission line parameters enables coherent conversion of a classical microwave input into a strongly correlated, fermionized Tonks–Girardeau gas of photons, characterized by pronounced antibunching and Friedel-type oscillations in the correlation functions (Fatis et al., 12 Jan 2026).

1. Theoretical Model and Effective Hamiltonian

Microwave photon propagation in a JJ transmission line of the type illustrated in [(Fatis et al., 12 Jan 2026), Fig. 1(a)] is modeled starting from a circuit Lagrangian. Each unit cell of length $a$ includes a capacitance to ground ($C_{0,n}$) and a series Josephson junction (capacitance $C_{J,n}$, critical current $I_{c,n}$). The Lagrangian reads: $$L = \frac12\sum_n C_{0,n}\dot\phi_n^2 +\frac12\sum_n C_{J,n}(\dot\phi_{n+1}-\dot\phi_n)^2 +\frac{\Phi_0}{2\pi}\sum_n I_{c,n}\cos\!\Bigl[\frac{2\pi}{\Phi_0}(\phi_{n+1}-\phi_n)\Bigr].$$ Expanding the cosine potential to quartic order and implementing the slowly-varying-envelope approximation (SVEA) yields, in the continuum limit, the action of a nonlinear Schrödinger field. Upon quantization, this leads to the Lieb–Liniger Hamiltonian for the envelope field $\hat\Psi(\zeta,\tau)$: $$\hat H = \int d\zeta \left\{ -\frac{\hbar^2}{2m} \hat\Psi^\dagger(\zeta)\frac{d^2}{d\zeta^2}\hat\Psi(\zeta) + \frac{g}{2}\hat\Psi^{\dagger2}(\zeta)\hat\Psi^2(\zeta) \right\}$$ where

• $m$ is an effective photon mass, set by the circuit parameters and frequencies,
• $g$ is the photon–photon interaction strength (derived from Josephson nonlinearity).

In terms of the underlying parameters:

• $C_0$: ground capacitance,
• $C_J$: JJ capacitance,
• $L_J = \Phi_0/(2\pi I_c)$: JJ inductance,
• $\omega_0$: carrier frequency,
• $\omega_P = 1/\sqrt{L_JC_J}$: plasma frequency,
• $c = C_J/C_0$.

Explicit expressions for $m$ and $g$: $$m = -\frac{\omega_P^2}{2}\frac{a\hbar}{\bar v_g^3 r} \frac{\left[-(1+4c)r^2+4c\right]^{3/2}(1-r^2)}{1+12c-\frac{2r^2}{1-r^2}}$$

$$g = -\left(\frac{2\pi}{\Phi_0}\right)^2 \frac{\hbar^2 \bar v_g^2 L_J}{2a} \frac{r^2}{\left[-(1+4c)r^2+4c\right](1-r^2)^2}$$

where $r = \omega_0/\omega_P$, and $\bar v_g$ is a reference group velocity.

2. Dimensionless Interaction Parameter and Quantum Regimes

The effective one-dimensional photon density under uniform stationary pump (photon flux $\Phi_{\rm ph}$) is: $\rho = \frac{\Phi_{\rm ph}}{\bar v_g}$ The interaction regime is quantified by the dimensionless Lieb–Liniger parameter: $$\gamma = \frac{m\,g}{\hbar^2\,\rho} = \frac{m\,g\,\bar v_g}{\hbar^2\,\Phi_{\rm ph}}$$ For $\gamma \ll 1$, the system is weakly interacting; for $\gamma \gg 1$, the repulsive interaction is so strong that the system enters the Tonks–Girardeau (TG) regime, wherein photons become effectively impenetrable.

Decreasing photon density (lower $\Phi_{\rm ph}$) or increasing nonlinearity (larger $g$) drives the system towards larger $\gamma$.

3. Adiabatic Tapering for Coherent State to TG Gas Conversion

A practical protocol for entering the TG regime relies on adiabatic spatial tapering of the transmission line. Specifically, one varies $L_J(z)$ and $C_J(z)$ smoothly along a macroscopic line length $L$, keeping $C_0$ and $\omega_P$ constant. This gradual change ensures that $\gamma(\tau)$, the dimensionless coupling at position $\tau = z/\bar v_g$, increases from $\gamma \ll 1$ at the input to $\gamma_{\rm final} \gg 1$ at the output.

To maintain adiabaticity in this gapless system, the rate of change must satisfy: $$|\partial_\tau \gamma(\tau)| \ll \Delta E_{\rm many-body}^2 / \hbar,$$ where the characteristic low-energy scale is $\Delta E_{\rm many-body} \sim \hbar^2 \rho^2 / m$. Numerical simulations with time-dependent tensor-network methods show that a tapering length $L$ in the range $500$ mm to $1$ m and a smooth profile are sufficient for adiabatic evolution to the TG ground state.

4. Tonks–Girardeau (Fermionized) Regime and Observables

In the TG limit $\gamma \to \infty$, strong repulsion makes the photon fluid impenetrable, mapping the Bose gas to a system of noninteracting fermions via the Girardeau transformation. The condition $\gamma \gg 1$, i.e., interaction energy per particle $g\rho$ greatly exceeding the kinetic energy ($\hbar^2\rho^2/{2m}$), is realized for state-of-the-art parameters, with $\gamma_{\rm final} \simeq 30\text{–}50$ at the line end.

A central observable is the zero-temperature second-order correlation function: $$g_2^{\rm TG}(\Delta t) = 1 - \left[\frac{\sin(\pi\,\Phi_{\rm ph}\,\Delta t)}{\pi\,\Phi_{\rm ph}\,\Delta t}\right]^2$$ This exhibits full antibunching ($g_2(0)=0$), and at larger $\Delta t$, damped Friedel oscillations at frequency $\pi\,\Phi_{\rm ph}$ set by the effective Fermi momentum. First-order coherence decays algebraically: $$g_1^{\rm TG}(\Delta t)\sim(\pi\,\Phi_{\rm ph}\,|\Delta t|)^{-1/2}, \qquad (\Delta t \to \infty)$$ In finite-length, adiabatically tapered lines, $g_2(\tau)$ transitions from Poissonian (coherent state) to the TG analytic form as the photon fluid develops strong correlations. Friedel oscillations become evident in $g_2$ for line lengths exceeding a few hundred millimeters.

5. Representative Circuit Parameters and Regime Accessibility

Typical parameters enabling access to the quantum fluid regime:

Parameter	Typical Value	Significance
$C_0$ (ground cap.)	30 fF	Sets background perm.
$C_J^\text{in}$	~15 fF	Tuned for input regime
$L_J^\text{in}$	~10 nH	JJ nonlinearity
$\omega_P$ / $2\pi$	5 GHz	JJ plasma frequency
$\omega_0$ / $2\pi$	4.15 GHz	Carrier frequency
$a$ (cell length)	10–20 $\mu$m	Defines spatial lattice

With these values:

• Group velocity: $v_g \simeq 0.1c$
• Photon–photon interaction: $g/\hbar \sim 10^5$ s${}^{-1} \mu$m${}^{-1}$
• Effective mass: $m \sim 10^{-35}$ kg
• Incident photon flux: $\Phi_{\rm ph} \simeq 7 \times 10^7$ s${}^{-1}$
• Photon density: $$\rho = \Phi_{\rm ph}/v_g \sim 3\, \mu\mathrm{m}^{-1}$$
• Resulting $\gamma \simeq 30$, placing the system deep in the TG regime

Under these experimental conditions, $g_2(\tau)$ measurements show near-perfect correspondence to the theoretical TG analytic predictions: antibunching $g_2(0)\approx 0$ and Friedel oscillations of amplitude $\sim 20\%$ with period $\tau_F = \Phi_{\rm ph}^{-1} \approx 14$ ns.

6. Measurement and Probing of Quantum Correlations

Quantum fluid behavior can be directly accessed by measuring the second-order coherence $g^{(2)}(\tau)$ using linear amplifiers and quadrature detection techniques. The emergence of antibunching and Friedel-type oscillations in $g^{(2)}(\tau)$ are robust, experimentally accessible signatures of fermionization in the photonic fluid. Time-resolved detection schemes allow observation of the crossover from coherent input statistics towards the highly correlated TG ground state (Fatis et al., 12 Jan 2026).

7. Significance and Perspectives

JJ transmission lines with engineered nonlinearity and adiabatic tapering emerge as a flexible and robust platform for simulating strongly correlated quantum fluids of light in a traveling-wave geometry. The ability to achieve and directly probe the Tonks–Girardeau regime in propagating microwave photons opens experimental access to new paradigms of many-body photonics, with high relevance for quantum simulation and quantum optics in the solid-state context. A plausible implication is the extension to more complex geometries or many-body Hamiltonians by circuit engineering and multiplexed drives, leveraging the high tunability and strong photon–photon interactions inherent to superconducting metamaterials (Fatis et al., 12 Jan 2026).